An optimal transport view of Schrödinger's equation

Abstract

We show that the Schrödinger equation is a lift of Newton's third law of motion {equation presented} on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric. Here the potential μ → F(μ) is the sum of the total classical potential energy hV, μi of the extended system and its Fisher information {equation presented}. The precise relation is established via a well-known (Madelung) transform which is shown to be a symplectic submersion of the standard symplectic structure of complex valued functions into the canonical symplectic space over the Wasserstein space. All computations are conducted in the framework of Otto's formal Riemannian calculus for optimal transportation of probability measures. © Canadian Mathematical Society 2011.